7. Given 5, 15, 45... if S = 1820, find n.
Answer (1)
Answer:Question:Given the geometric sequence 5, 15, 45,... if the sum \(S_n = 1820\), find n.Solution Process:The given sequence is a geometric sequence with the first term \(a = 5\) and the common ratio \(r = \frac{15}{5} = 3\).The sum of the first n terms of a geometric sequence is given by the formula:\(S_n = \frac{a(r^n - 1)}{r - 1}\)We are given that \(S_n = 1820\). Substituting the values of a and r into the formula, we get:\(1820 = \frac{5(3^n - 1)}{3 - 1}\)\(1820 = \frac{5(3^n - 1)}{2}\)Multiply both sides by 2:\(3640 = 5(3^n - 1)\)Divide both sides by 5:\(728 = 3^n - 1\)Add 1 to both sides:\(729 = 3^n\)We need to find the value of n such that \(3^n = 729\). We can express 729 as a power of 3:\(729 = 3^6\)Therefore, \(3^n = 3^6\), which implies \(n = 6\).Answer:\(n = \boxed{6}\)Step-by-step explanation:
